Integral homology 3-spheres and the Johnson filtration

Pitsch, Wolfgang

doi:10.1090/S0002-9947-08-04208-6

Integral homology -spheres and the Johnson filtration

Author: Wolfgang Pitsch
Journal: Trans. Amer. Math. Soc. 360 (2008), 2825-2847
MSC (2000): Primary 57M99; Secondary 20F38, 20F12
DOI: https://doi.org/10.1090/S0002-9947-08-04208-6
Published electronically: January 4, 2008
MathSciNet review: 2379777
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Abstract: The mapping class group of an oriented surface $\Sigma_{g,1}$ of genus with one boundary component has a natural decreasing filtration $\mathcal{M}_{g,1} \supset \mathcal{M}_{g,1}(1) \supset \mathcal{M}_{g,1}(2) \supset \mathcal{M}_{g,1}(3) \supset \cdots$ , where $\mathcal{M}_{g,1}(k)$ is the kernel of the action of $\mathcal{M}_{g,1}$ on the $k^{th}$ nilpotent quotient of $\pi_1(\Sigma_{g,1})$ . Using a tree Lie algebra approximating the graded Lie algebra $\displaystyle \bigoplus_{k} \mathcal{M}_{g,1}(k)/\mathcal{M}_{g,1}(k+1)$ we prove that any integral homology sphere of dimension has for some a Heegaard decomposition of the form $M = \mathcal{H}_g \coprod_{\iota_g \phi} - \mathcal{H}_g$ , where $\phi \in \mathcal{M}_{g,1}(3)$ and $\iota_g$ is such that $\mathcal{H}_g \coprod_{\iota_g} - \mathcal{H}_g= S^3$ . This proves a conjecture due to S. Morita and shows that the ``core'' of the Casson invariant is indeed the Casson invariant.

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